A family of non-injective skinning maps with critical points

Abstract

Certain classes of 3-manifolds, following Thurston, give rise to a ‘skinning map’, a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold M M , a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M M to conclude that the skinning map associated to M M sends a specified path to itself and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. A family of finite covers of M M produces examples of non-immersion skinning maps on the Teichmüller spaces of surfaces in each even genus, and with either 4 4 or 6 6 punctures.